Dec 29, 2014 - expected to be the pseudo-Goldstone acoustic mode of bro- ...... 134 M. Y. Veillette, A. J. A. James, and F. H. L. Essler. 'Spin dy- namics of the ...
Unusual ordered phases of highly frustrated magnets: a review Oleg A. Starykh1 1
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112-0830 (Dated: December 29, 2014)
arXiv:1412.8482v1 [cond-mat.str-el] 29 Dec 2014
We review ground states and excitations of a quantum antiferromagnet on triangular and other frustrated lattices. We pay special attention to the combined effects of magnetic field h, spatial anisotropy R, and spin magnitude S. The focus of the review is on the novel collinear spin density wave and spin nematic states, which are characterized by fully gapped transverse spin excitations with S z = ±1. We discuss extensively R − h phase diagram of the antiferromagnet, both in the large-S semiclassical limit and the quantum S = 1/2 limit. When possible, we point out connections with experimental findings. [This is the originally submitted version of the invited review, to be published in Reports on Progress in Physics in 2015. The link, via DOI, to the accepted and published version of the manuscript, which is updated according to the referees comments, will be provided upon its actual publication.]
Contents
I. Introduction
1
II. Classical model in a magnetic field
2
III. Quantum model in magnetic field A. Isotropic triangular lattice B. Spatially anisotropic triangular antiferromagnet with J 0 6= J C. Spin 1/2 spatially anisotropic triangular antiferromagnet with J 0 6= J
4 4
IV. SDW and nematic phases of spin-1/2 models A. SDW 1. SDW in a system of Ising-like coupled chains 2. Magnetization plateau as a commensurate collinear SDW phase B. Spin nematic 1. Weakly coupled nematic chains 2. Spin-current nematic state at the 1/3-magnetization plateau C. Magnetization plateaus in itinerant electron systems
9 9 10
V. Experiments A. Magnetization plateau B. SDW and spin nematic phases C. Weak Mott insulators: Hubbard model on anisotropic triangular lattice
5 8
11 11 11 12 12 13 14 15 15
Acknowledgments
16
References
16
I.
INTRODUCTION
Frustrated quantum antiferromagnets have been at the center of intense experimental and theoretical investigations for many years. These relentless efforts have very recently resulted in a number of theoretical and experimental breakthroughs: quantum entanglement1–3 , density matrix renormal-
ization group (DMRG) revolution and Z2 liquids in kagom´e and J1 −J2 square lattice models4–6 , spin-liquid-like behavior in organic Mott insulators7,8 and kagom´e lattice antiferromagnet herbertsmithite9 . Along the way, a large number of frustrated insulating magnetic materials featuring rather unusual ordered phases, such as magnetization plateaux, longitudinal spin-density waves, and spin nematics, has been discovered and studied. It is these ordered, yet sufficiently unconventional, states of magnetic matter and theoretical models motivated by them that are the subject of this Key Issue article. This review focuses on materials and models based on simple triangular lattice, which, despite many years of fruitful research, continue to supply us with novel quantum states and phenomena. Triangular lattice represents, perhaps, the most widely studied frustrated geometry10–12 . Indeed, the Ising antiferromagnet on the triangular lattice was the first spin model found to possess a disordered ground state and extensive residual entropy13 at zero temperature. While the classical Heisenberg model on the triangular lattice does order at T = 0 into a well-known 120◦ commensurate √ √ spiral pattern (also known as a three-sublattice or 3 × 3 state), the fate of the quantum spin-1/2 Heisenberg Hamiltonian has been the subject of a long and fruitful debate spanning over 30 years of research. Eventually it was firmly established that the quantum spin-1/2 model remains ordered in the classical 120◦ pattern14–16 . Although the originally proposed resonating valence bond liquid17,18 did not emerge as the ground state of the spin-1/2 Heisenberg model, such a phase was later found in a related quantum dimer model on the triangular lattice.19 It turns out that a simple generalization of the triangular lattice Heisenberg model whereby exchange interactions on the nearest-neighbor bonds of the triangular lattice take two different values – J on the horizontal bonds and J 0 on the diagonal bonds, as shown in Figure 1 – leads to a very rich and not yet fully understood phase diagram which sensitively depends on the magnitude of the site spin S and magnetic field h. Such a distorted, or spatially anisotropic, triangular lattice model interpolates between simple unfrustrated square lattice (J = 0, J 0 6= 0), strongly frustrated triangular lattice (J = J 0 6= 0) and decoupled spin chains (J 6= 0, J 0 = 0). An unexpectedly large number of experimental systems
2 Heisenberg spins with isotropic exchange interactions exhibit a large accidental classical ground-state degeneracy. That is, at finite magnetic fields, there exists an infinite number of continuously deformable classical spin configurations that constitute minimum energy states, but are not symmetry related. This degeneracy is understood by the observation that the Hamiltonian of the isotropic triangular lattice antiferromagnet in magnetic field h can be written, up to an unessential constant, as
FIG. 1: Deformed triangular lattice. Solid (thin) lines denote bonds with exchange constant J (J 0 ) correspondingly. Also indicated are nearest-neighbor vectors δj as well as A, B and C sublattice structure.
seems to fit into this simple distorted triangular lattice model with spin S = 1/2: Cs2 CuCl4 (shows extended spinon continuum, J 0 /J = 0.34), Cs2 CuBr4 (shows magnetization plateau, J 0 /J ≈ 0.7) and Ba3 CoSb2 O9 (shows magnetization plateau, J 0 /J ≈ 1). Importantly, a number of very interesting organic Mott insulators of X[Pd(dmit)2 ]2 and κ-(ET)2 Z families can also be approximately described by the spatially anisotropic J − J 0 model with additional ring-exchange (involving four spins and higher order) terms. It is widely believed that these materials are weak Mott insulators in a sense of being close to a metal-insulator transition. As such, they are best described by distorted t − t0 − U Hubbard model, so that spatial anisotropy of exchanges follows from that of single particle hopping parameters, J 0 /J ∼ (t0 /t)2 . The (intentionally) rather narrow focus of the review leaves out several important recent developments. Kagom´e lattice antiferromagnets represent perhaps the most notable omission. A lot is happening there, both in terms of experiments on materials such as herbertsmithite and volborthite20 and in terms of theoretical developments4,9 . This very important area of frustrated magnetism deserves its own review. We also have avoided another very significant area of development – systems with significant spin-orbit interactions. Progress in this area has been summarized in recent reviews21,22 . The presentation is organized as follows: Section II contains brief review of the states of classical model in a magnetic field. Section III describes phase diagrams of the semiclassical S 1 (Section III B) and S = 1/2 (Section III C) models. Novel ordered states, a collinear SDW and a spin nematic, which are characterized by the absence of S = 1 transverse spin excitations, are described in Section IV. Section V summarizes key experimental findings relevant to the review, including recent developments in organic Mott insulators.
II.
CLASSICAL MODEL IN A MAGNETIC FIELD
Triangular antiferromagnets in an external magnetic field have been extensively studied for decades, and found to possess unusual magnetization physics that remains only partially understood. Underlying much of this interesting behavior is the discovery, made long ago,23 that in a magnetic field,
H0 =
J Xh h i2 Sr + Sr+δ1 + Sr+δ2 − . 2 r 3J
(1)
The sum is over all sites r of the lattice and nearestneighbor vectors δ1,2 are indicated in the Figure 1. Note that Zeeman terms Sr · h appear three times for every spin in this sum, which explains the factor of 1/3 in the h term in (1). We immediately observe that every spin configuration which nullifies every term in the sum (1) belongs to the lowest energy manifold of the model. Given the side-sharing property of the triangular lattice, so that fixing all spins in one elementary triangle fixes two spins in each of the adjacent triangles, sharing sides with the first one, this implies that all such states exhibit a three-sublattice structure and must satisfy SA + SB + SC =
h . 3J
(2)
This condition provides 3 equations for 6 angles needed to describe 3 classical unit vectors. In the absence of the field, the 3 undetermined angles can be thought of as Euler’s angles of the plane in which the spins spontaneously form a threesublattice 120◦ structure. However, this remarkable feature persists for h 6= 0 as well. There, the symmetry of the Hamiltonian (1) is reduced to U (1) but the degeneracy persists: one of the free angles can be thought as gauge degree of freedom to rotate all spins about the axis of the field h, while the remaining two constitute the phenomenon of accidental degeneracy. Remarkably, thermal (entropic) fluctuations lift this extensive degeneracy in favor of the two coplanar (Y and V states) and one collinear (UUD) spin configurations, shown in Figure 2. Symmetry-wise, coplanar states break two different symmetries – a discrete Z3 symmetry, which corresponds to the choice of sublattice on which the down spin (in the case of Y) or the minority spin (in the case of V) is located, and a continuous U (1) = O(2) symmetry of rotations about the field axis. The collinear UUD state breaks only the discrete Z3 symmetry (a choice of sublattice for the down spin). Selection of these simple states out of infinitely many configurations, which satisfy (2), by thermal fluctuations represents a textbook example of the ‘order-by-disorder’ phenomenon24 . The resulting enigmatic phase diagram, first sketched by Kawamura and Miyashita in 1985, Ref. 23, continues to attract much attention - and in fact remains not fully understood. Figure 3 shows the result of recent simulations25 , which studied critical properties of various phase transitions in great detail, differs in important aspect from the original suggestion –
3 very large) below Tv , resulting in a disordered “spin-gel” state31 . Whether the change from high-temperature vortexdominated regime to the low-temperature spin-fluctuationdominated one is a true transition or a sharp crossover remains the topic of active debate32–36 . Experimental ramifications of this interesting scenario are reviewed in37 .
FIG. 2: Various spin configurations from the classical ground state manifold. (a) coplanar Y state, (b) collinear UUD (up-up-down), (c) coplanar V state, (d) non-coplanar cone (umbrella) state, (e) inverted Y state.
it is established now that there is no direct transition between the Y state and the paramagnetic phase (very similar results were obtained in an extensive study26 ). The two phases are separated by the intervening UUD state which extends down to lowest accessible field values and before the transition to the paramagnetic state. Figure 3 also shows that of all entropically selected state, the UUD state is most stable - it extends to higher T than either Y or V states. The UUD is also the most ‘visible’ of the three selected states - it shows up as a plateau-like feature in the magnetization curve M (h), see Figure 4 which shows experimental data for a S = 5/2 triangular lattice antiferromagnet RbFe(MoO4 )2 27 . Notice that a strict magnetization plateau at 1/3 of the full (saturation) value Msat , M = Msat /3, is possible only in the quantum problem (i.e. the problem with finite spin S), when all spin-changing excitations with S z 6= 0 are characterized by gapped spectra, and at absolute zero T = 0, when no excitations are present in the ground state (see next Section III for complete discussion). At any finite T thermally excited spin waves are present and lead to a finite, albeit different from the neighboring non-plateau states, slope of the magnetization M (h). In the case of the classical problem we review here the gap in spin excitation spectra is itself T -dependent and disappears as T → 0: as a result its magnetization ‘quasi-plateau’ too disappears in the T → 0 limit. However, at any finite T , less than that of the transition to the paramagnetic state, the slope of the M (h) for the UUD state is different from that of the Y and V states28 . The most outstanding, and still not resolved issue, is the conjectured SO(3) breaking transition29 at Tv ≈ 0.285J and h = 0. The transition is driven by the proliferation, above the critical temperature Tv , of the Z2 vortices, which are defects of spin chirality29 . It is by now established that unlike the case of Berezinsky-Kosterlitz-Thouless vortex-unbinding transition30 , the spin correlation length remains finite (albeit
FIG. 3: Magnetic field phase diagram of the classical triangular lattice antiferromagnet. [Adapted from Seabra et al., Phys. Rev. B 84, 214418 (2011). Copyright 2011 by the American Physical Society.] Transition points determined by the Monte Carlo simulations are shown by filled symbols. Continuous phase transitions are drawn with a dashed line, while Berezinskii-Kosterlitz-Thouless phase transitions are drawn with a dotted line. For fields h ≤ 3 a double transition is found upon cooling from the paramagnet, while for h ≥ 3 only a single transition is found. Behavior of the phase transition lines in the low-field region h ≤ 0.2, which is left unshaded in the diagram, is not settled at the present. See Ref.31 for the recent study of h = 0 line.
A classical system with spatially anisotropic interactions offers an interesting generalization of the ‘order-by-disorder’ phenomenon. Consider slightly deformed triangular lattice, with J 0 < J (see Fig. 1). An arbitrary weak deformation lifts, at T = 0, the accidental degeneracy in favor of the simple non-coplanar umbrella state (configuration ‘d’ in Fig. 2) in the whole range of h below the saturation field. The energy gain of this well-known spin configuration is of the order (J − J 0 )2 /J. One thus can expect that, for sufficiently small difference R = J − J 0 , entropic fluctuations, which favor coplanar and collinear spin states at a finite T , can still overcome this classical energy gain and stabilize collinear and coplanar states above some critical temperature which can be estimated as Tumb−uud ∼ R2 /J. As a result, the UUD phase ‘decouples’ from the T = 0 axis. The leftmost point of the UUD phase in the h − T phase diagram now occurs at a finite Tumb−uud as was indeed observed in Monte Carlo simulation of the simple triangular lattice model in Ref.28 (see Fig.10
4 of that reference), as well as in a more complicated ones, for models defined on deformed pyrochlore38 and ShastrySutherland39 lattices. It is worth noting that the roots of this behavior can be traced to the famous Pomeranchuk effect in 3 He, where the crystal phase has higher entropy than the normal Fermi-liquid phase. As a result, upon heating, the liquid phase freezes into a solid40,41 . In the present classical spin problem it is ‘superfluid’ umbrella phase which freezes into a ‘solid’ UUD phase upon heating (see discussion of the relevant terminology at the end of Section III A below).
FIG. 4: Magnetization curve M (h) of a spin-5/2 antiferromagnet RbFe(MoO4 )2 at T = 1.3K. [Adapted from Smirnov et al., Phys. Rev. B 75, 134412 (2007). Copyright 2007 by the American Physical Society.]
III.
QUANTUM MODEL IN MAGNETIC FIELD A.
Isotropic triangular lattice
Much of the intuition about selection of coplanar and collinear spin states by thermal fluctuations applies to the most interesting case of quantum spin model on a (deformed) triangular lattice. Only now it is quantum fluctuations (zero-point motion) which differentiate between different classicallydegenerate (or, nearly degenerate) states and lift the accidental degeneracy. One of the first explicit calculations of this effect was carried out by Chubukov and Golosov42 , who used semiclassical large-S spin wave expansion in order to systematically separate classical and quantum effects. This well-known technique relies on Holstein-Primakoff representation of spin operators in terms of bosons. The representation is nonlinear, Srz = S − a†r ar , Sr+ = (2S − a†r ar )1/2 ar , and leads, upon expansion of square roots in powers of small P parameter 1/S, to ∞ bosonic spin-wave Hamiltonian H = Ecl + k=2 H (k) . Here Ecl stands for the classical energy of spin configuration, which scales as S 2 , while each of the subsequent terms H (k) are of k-th order in operators ar and scale as S 2−k/2 . Diagonalization of quadratic term H (2) provides one with the dispersion (m) ωk of spin wave excitations (k is the wave vector and m is the band index), in terms of which quantum zero-point energy
P (m) is given by hH (2) i = (1/2) m,k ωk . This energy scales as S, and thus provides the leading quantum correction to the classical (∝ S 2 ) result. The main outcome of the calculation42 is the finding that quantum fluctuations too selects coplanar Y and V and collinear UUD states (states a, b and c in Figure 2) out of many other classically degenerate ones. The authors also recognized the key special feature of the UUD state - being collinear, this state preserves U (1) symmetry of rotations about the magnetic field axis. The absence of broken continuous symmetry implies that spin excitations have a finite energy gap in the dispersion. This expectation is fully confirmed by the explicit calculation42 which finds that the gaps of the two low energy modes are given by |h − h0c1,c2 |, where the lower/upper critical fields are given by h0c1 = 3JS − 0.5JS/(2S) and h0c2 = 3JS + 1.3JS/(2S), correspondingly. (The third mode describes a high-energy precession with energy hS.) The uniform magnetization M , being the integral of motion, remains at 1/3 of the maximum (saturation) value, M = Msat /3, in the UUD stability interval h0c1 < h < h0c2 . As a result, magnetization curve M (h) of the quantum triangular lattice antiferromagnet is non-monotonic and exhibit striking 1/3 magnetization plateau in the finite field interval ∆h = h0c2 − h0c1 = 1.8JS/(2S). It is worth noting that this is not a narrow interval at all, ∆h/hsat = 0.2/(2S) in terms of the saturation field hsat = 9JS. Extending this large-S result all way to the S = 1/2 implies that the magnetization plateau takes up 20% of the whole magnetic field interval 0 < h < hsat . Numerical studies of the plateau focus mostly on the quantum spin-1/2 problem (numerical studies of higher spins are much more ‘expensive’) and confirm the scenario outlined above. The plateau at M = Msat /3 is indeed found, and moreover its width in magnetic field agrees very well with the described above large-S result, extrapolated to S = 1/2.43–48 The pattern of symmetry breaking by the coplanar/collinear states is described by the following spin expectation values Y state, UUD state, V state,
0 < h < h0c1 : hSr+ i = aeiϕ sin[Q · r], hSrz i = b − c cos2 [Q · r]; (3) 0 0 + hc1 < h < hc2 : hSr i = 0, hSrz i = M − c cos[Q · r]; (4) 0 + iϕ hc1 < h < hsat : hSr i = ae cos[Q · r], hSrz i = b − c cos2 [Q · r]. (5)
Here the ordering wave vector Q = (4π/3, 0) is commensurate with the lattice which results in only three possible values that the product Q · r = 2πν/3 can take (ν = 0, 1, 2), modulo 2π. Angle ϕ specifies orientation of the ordering plane for coplanar spin configurations within the x − y plane, M is magnetization per site, and parameters a, b, c are constants dependent upon the field magnitude. Connection with ‘super’ phases of bosons: Eq.(4) identifies UUD state as a collinear ordered state which can be thought of as solid. Its ‘density’ hSrz i is modulated as cos[Q · r], as appropriate for the solid, and as a result its local magnetization follows simple ‘up-up-down’ pattern within each
5 elementary triangle. It obviously respects U (1) symmetry of rotations about S z axis. The coplanar Y and V states break this U (1) symmetry by spontaneously selecting angle ϕ. Note that in addition they are characterized by the modulated density hSrz i, which makes them supersolids: the superfluid order (magnetic order in the x − y plane as selected by ϕ) co-exists with the solid one (modulated z component, or density). This useful connection is easiest to make precise49 in the case of S = 1/2 when the following mapping between a hard-core lattice Bose gas an a spin-1/2 quantum magnet can easily be established: + − + − a+ r ↔ Sr , ar ↔ Sr , nr = ar ar ↔ Sr − 1/2.
used as a convenient starting point for accessing more complicated states. This approach, which amounts to the investigation of the local stability of the UUD phase, was carried out in50 . Similar to42 , the calculation is based on the three-spin UUD unit cell, resulting in three spin wave branches. One of these, describing total spin precession, is a high-energy mode not essential for our analysis, while the two others, which describe relative fluctuations of spins, are the relevant lowenergy modes.
(6)
The superfluid order is associated with finite ha+ r i while the solid one with modulated (with momentum Q in our notations) boson density hnr i.
B.
Spatially anisotropic triangular antiferromagnet with J 0 6= J
Consider now a simple deformation of the triangular lattice which makes exchange interaction on diagonal bonds, J 0 , different from those on horizontal ones J, so that R = J − J 0 6= 0. This simple generalization of the Heisenberg model leads to surprisingly complicated and not yet fully understood phase diagram in the magnetic field (h) - deformation (R) plane. Semiclassical (S 1) analysis of this problem is complicated by the fact that arbitrary small R 6= 0 removes accidental degeneracy of the problem in favor of the unique noncoplanar and incommensurate cone (umbrella) state, already discussed in Sec. II, state ‘d’ in Fig. 2. This simple state gains energy of the order δEclass ∼ S 2 R2 /J per spin. Its structure is described by hSr i = M zˆ + c(cos[Q0 · r + ϕ]ˆ x + sin[Q0 · r + ϕ]ˆ y ), (7) where classically the ordering wave vector Q0 = (2 cos−1 [−J 0 /2J], 0) is a continuous function of J 0 /J. Being non-coplanar, this state is characterized by the finite chirality χ ∼ Sr · Sr+δ1 × Sr+δ2 ∼ M (2 sin[Q0 · δ1 ] − sin[Q0 · δ2 ]). At the same time, for sufficiently small R, the quantum energy gain due to zero-point motion of spins, which is of the order δEq ∼ SJ per spin, should be able to overcome δEclass and still stabilize one of the coplanar/collinear states considered in Sec. III A above. Comparing the two contributions, we conclude, following Ref. 50, that the classical-quantum competition can be parameterized by the dimensionless parameter δ ∼ δEclass /δEq ∼ S(R/J)2 . (In the following, we will use more precise value δ = (40/3)S(R/J)2 , with numerical factor 40/3 as introduced in50 for technical convenience.) Explicit consideration of this competition is rather difficult due to complicated dependence of the parameters of the coplanar states on magnetic field h and exchange deformation R. However, inside the M = Msat /3 magnetization plateau phase, the spin structure is actually pretty simple, as equation (4) shows, which suggests that UUD state can be
FIG. 5: Schematic large-S phase diagram of the deformed triangular lattice antiferromagnet. Phase boundary lines are schematic and serve to indicate possible shape only approximately. Dashed lines indicate location of conjectured phase transitions. Capital letters A through O indicated various (multi)critical points discussed in the text. Each phase is characterized by the set of broken symmetries, as indicated by (blue) symbols. Observe that the line H-B-b-C2 -C-C1 a-A-O is the transition line separating phases with broken Z3 from those with broken O(2). This phase diagram is based on Refs. 50–52.
Within the UUD plateau phase, which is bounded by FACBG lines in Figure 5, both low energy spin wave modes remain gapped. As discussed above, the only symmetry this collinear state breaks is Z3 . The gaps of the low-energy spin wave modes (to be called mode 1 and mode 2 in the following) are given by |h − hc1,c2 |, where now the lower and upper critical fields hc1,c2 (δ) depend on the dimensionless deformation parameter δ. Gap’s closing at the lower hc1 (upper hc2 ) critical fields of the plateau implies Bose-Einstein condensation (BEC) of the appropriate magnon mode (1 or 2) and the appearance of the transverse to the field spin component hSr+ i = 6 0, see (3) and (5). This BEC transition breaks spin rotational U (1) symmetry via spontaneous selection of the ‘superfluid’ phase ϕ. The spin structure of the resulting ‘condensed’ state sensitively depends on the wave vector k1/2 of the condensed magnon. Key results of the large-S calculation in Refs. 50–52 can now be summarized as follows: (a) In the interval 0 < δ ≤ 1 the lower critical field is actually
6 independent of δ, hc1 = h0c1 , and the minimum of the spin wave mode 1 remains at k1 = 0. As a result, BEC condensation of mode 1 at h = hc1 signals the transition to the commensurate Y state, which lives in the region OAF in Figure 5. In addition to breaking the continuous U (1) symmetry, the Y state inherits broken Z3 from the UUD phase (which corresponds to the selection of the sublattice for the down spins). (b) For 1 < δ ≤ 4, the low critical field increases and, simultaneously, the spin wave minima shift from zero to finite momenta ±k1 = (±k1 , 0). Thus, at h = hc1 , the spectrum softens at two different wave vectors at the same time. This opens an interesting possibility of the simultaneous coherent condensation of magnons with opposite momenta +k1 and −k1 .53 It turns out, however, that repulsive interaction between condensate densities at ±k1 makes this energetically unfavorable50 . Instead, at the transition the symmetry between the two possible condensates is broken spontaneously and magnons condense at a single momentum, +k1 or −k1 . The resulting state, denoted as distorted umbrella in50 , is characterized by the broken Z3 , U (1) and Z2 symmetries – the latter corresponds to the choice +k1 or −k1 . This non-coplanar state lives in the narrow region bounded by lines AC1 (solid) and AaC1 (dashed) in Figure 5. (c) Similar developments occur near the upper critical field hc2 . In the interval 0 < δ ≤ 3 the upper critical field is actually independent of δ, hc2 = h0c2 , and the minimum of the spin wave mode 2 remains at k2 = 0. The transition on the line GB is towards the coplanar and commensurate V state, bounded by GBH in Figure 5. This state is characterized by broken Z3 × U (1) symmetries. (d) In the interval 3 < δ ≤ 4, the upper field hc2 diminishes and simultaneously the minimum of the mode 2 shifts from zero momentum to the two degenerate locations at ±k2 = (±k2 , 0). Here, too, at the condensation transition (along the line BC2 ) it is energetically preferable to break Z2 symmetry between the two condensates and to spontaneously select just one momentum, +k2 or −k2 . This leads to distorted umbrella with broken Z3 × U (1) × Z2 symmetries. This state lives between (solid) BC2 and (dashed) BbC2 lines in Figure 5. (e) Spin nematic region. The critical fields hc1 and hc2 merge at the plateau’s end-point δ = 4 (point C). The minima of the spin wave modes 1 p and 2 coincide at this point k1 = k2 = (k0 , 0) with k0 = 3/(10S). The end-point of the UUD phase thus emerges as a point of an extended symmetry hosting four linearly-dispersing gapless spin wave modes50 (two branches, each gapless at (±k0 , 0)). Single-particle analysis of possible instabilities at the plateau’s end-point (point C, δ = 4) shows that in addition to the expected U (1) × U (1) symmetry (the two U (1)’s represent phases of the single particle condensates), the state at δ = 4 point posses an unusual P1 symmetry – the magnitude of the condensate at this point is not constrained50 . The enhanced degeneracy of the plateau’s end point C can also be understood from the observation that the two chiral distorted umbrella states merging at the point C are characterized, quite generally, by the different chiralities: Condensation of magnons at hc1,c2 , described in items (b) and (c) above, proceed independently of each other - hence the
chiralities of the two states are not related in any way. Thus the merging point of the two phases, point C, must possess enhanced symmetry. Unconstrained magnitude mode hints at a possibility of a two-particle condensation - and indeed recent work51 has found that a single-particle instability at δ = 4 is pre-empted by the two-particle one at a slightly smaller value of δ = δcr = 4 − O(1/S 2 ). This is indicated by the (dashed) line C1 -C2 in the Figure. This happens via the development of the ‘superconducting’-like instability of the magnon pair fields Ψ1,p = d1,+k0 +p d2,−k0 −p and Ψ2,p = d1,−k0 +p d2,+k0 −p and consists in the appearance of two-particle condensates hΨ1,p i = hΨ2,p i = iΥ/|p|. The sign of the real-valued Ising order parameter Υ determines the sense of spin-current circulation on the links of the triangular lattice, as illustrated in Figure 6. The spin current is defined as the ground state expectation value of the vector product of neighboring spins. For example, spin current on the AC link of the elementary trianz gle is given by JAC = zˆ · hSA × SC i ∼ Υ. In addition to spin currents, this novel state also supports finite spin chirality, x,y i = 0 for each of the hSA ·SB ×Si ∼ Υ, even though hSA/B/C spins individually. At the same time, in the absence of singleparticle condensation, hd1/2 i = 0, the usual two-point spin correlation function hS a (r)S b (0)i is not affected by the twoparticle hΨ1/2 i 6= 0 condensate: its transverse (a = b = x or y) components continue to decay exponentially because of the finite energy gap in the single magnon spectra, while the longitudinal (a = b = z) components continue to show a perfect UUD crystal order. Hence the resulting state, which lives inside triangle-shaped region C1 -C-C2 in Figure 5, is a spin-nematic state. It can also be called a spin-current state51 . It is uniquely characterized by the sign of Υ which determines the sense (clockwise or counterclockwise) of spin current circulation. The spin-current nematic is an Ising-like phase with massive excitations, which are domain walls separating domains of oppositely circling spin currents. It is characterized by broken Z3 ×Z2 , where Z2 is the sign of the two-magnon order parameter Υ in the ground state. It is useful to note that spontaneous selection of the circulation direction can also be viewed as a spontaneous breaking of the spatial inversion symmetry I : x → −x, which changes direction of spin currents on all bonds. (A different kind of nematic is discussed in Section IV B.)
FIG. 6: Pattern of spin currents in the spin nematic phase, for Υ > 0.
(f) high-field region, hc2 (δ) h ≤ hsat (δ). The high field region, h ≈ hsat (δ), can be conveniently analyzed within powerful Bose-Einstein condensation (BEC) framework. This fol-
7 lows from the simple observation that the ground state of the spin-S quantum model becomes a simple fully polarized state once the magnetic field is greater than the saturation field, h > hsat (δ) (which itself is a function of spin S and exchange deformation R). Excitations above this exact ground state are standard spin waves minimal energy for creating which is given by h − hsat . Similar to the situation near plateau’s critical field hc1/c2 , these spin waves are characterized by nontrivial dispersion with two degenerate minima at momenta ±Q0 (see expression below (7)). Spontaneous condensation in one of the minima, which constitutes breaking of Z2 symmetry, results in the usual cone, or umbrella, state which is characterized by the spin pattern (7) which breaks spin-rotational O(2). This state is realized to the right of the D-C2 line in Figure 5. Instead, simultaneous condensation of the high-field magnons in both minima results in the coplanar state53 (also known as fan state) of the V type. For any δ 6= 0 the wave vector Q0 is not commensurate with the lattice, which makes the coplanar state to be incommensurate as well. The condensates √ Ψ = ρeiθ1,2 at wave vectors ±Q0 have equal magnitude √1,2 ρ and each breaks U (1) symmetry. By choosing the phases θ1,2 , the resulting state breaks two O(2) symmetries. In total, the coplanar state is characterized by the broken O(2) × O(2). It is instructive to think of these symmetries in a slightly different way - as of those associated with the total θ+ = θ1 + θ2 and the relative θ− = θ1 − θ2 phases. The spin structure of this state is described by the incommensurate version of (5) which can be obtained by identifying ϕ = θ+ and replacing cos[Q · r] → cos[Q0 · r + θ− ]. The latter replacement shows that the relative O(2) symmetry, associated with the phase θ− , can also be thought of as a translational symmetry associated with the shift of the spin configuration by the vector r0 such that Q0 · r0 = θ− , modulo 2π. Moving to the left, we come to the special commensurate point, δ = 0 at h = hsat . Here Q0 → Q = (4π/3, 0), resulting in the commensurate coplanar V state. As noted right below Eq. (5), here Q · r = 2πν/3, with ν = 0, 1, 2, making the V state a three-sublattice one. Continuous O(2) translational symmetry of the incommensurate V phase is replaced here by the discrete Z3 symmetry. In other words, the relative phase θ− is now restricted to the set of three equivalent (up to a global translation of the lattice) degenerate values. The Z3 → O(2) transition across the line H-B between the two coplanar states is thus of a commensurate-incommensurate transition (CI) type. It is described by the classical twodimensional sine-Gordon model with the nonlinear cos[3θ− ] potential describing the locking of the relative phase to the Z3 set.47 In the vicinity of point H in√ the diagram the line of the CI transition follows hsat − h ∼ δ.52 While the arguments presented here are valid in the immediate vicinity of hsat , the identification of the whole line H-B as the CIT line between the commensurate and incommensurate V states is possible due to the additional evidence reported in item (c) above – commensurate V state is reached from the UUD state in the whole interval 0 < δ ≤ 3. The next task is to connect the incommensurate coplanar V state, which occupies region HDB in Figure 5, with the in-
commensurate cone state, to the right of D-C2 line. Since the V state has equal densities of bosons in the ±Q0 points, while the cone has finite density only in one of them, continuous transition between these two states at finite condensate density (that is, at any h < hsat ) is not possible. At infinitesimally small condensate density, i.e. at h = hsat , direct transition is possible - it occurs at point D, which is a point of extended O(2) × O(2) × O(2) symmetry: the two O(2)’s are phase symmetries while the third one is an emergent symmetry associated with the invariance of the potential energy at the constant total condensate density, ρ = ρ1 + ρ2 , with respect to the distribution of condensate densities ρj=1,2 = Ψ†j Ψj between the ±Q0 momenta. Large-S calculation of ladder diagrams which describe quantum corrections to the condensate energy place point D at δ = 2.9152 . Assuming that the first order transition does not realize, we are forced to conclude that V and cone states must be separated by the intermediate phase, occupying DBbC2 region. This phase breaks the O(2) symmetry between ρ1 and ρ2 and interpolates smoothly between the symmetric situation ρ1 = ρ2 , on the D-B line, and the asymmetric one ρ = ρ1 and ρ2 = 0 (or vice versa) on the line D-C2 . In doing so the momentum of the ‘minority’ condensate is found to evolve continuously from the initial q2 = Q0 (which coincides with the momentum of the ‘majority’ condensate ρ1 ) on the line D-C2 to the final q2 = −Q0 on the D-B line52 . The resulting phase is a non-coplanar one, with strongly pronounced asymmetry in the x − y plane: 0 √ √ hSr+ i = ρ1 eiθ1 eiQ ·r + ρ2 eiθ2 eiq2 ·r . The state is characterized by the broken O(2) × O(2) × Z2 . For the lack of better term we call it double spiral52 . Going down along the field axis takes us toward the dashed B-b-C2 line below which, according to the analysis summarized in item (c) above, represents a phase with broken Z3 × Z2 × O(2). Hence along this line Z3 is replaced by O(2), which makes it a continuation of the C-IC transitions line H-B. (g) low-field region, 0 ≤ h hc1 (δ). Semiclassical analysis at zero field h = 0 is well established and predicts incommensurate spiral state with zero total magnetization M = 0 of course. Quantum fluctuations renormalize strongly parameters of the spin spiral54 . The most quantum case of the spin S = 1/2 remains not fully understood even for the relatively weak deformation of exchanges R = J − J 0 ≤ J and is described in more details in Section III C. At δ = 0 one again has commensurate three-sublattice antiferromagnetic state, widely known as a 120◦ structure, which evolves into commensurate Y state in external magnetic field h 6= 0. Phenomenological analysis of Ref.47, Section III E, shows Y state becomes incommensurate when the deformation R exceeds Rc ∼ h3/2 . Analysis near hc1 , reported in (b), tells that C-IC transition line must end up at point A. Comparing the energies of the incommensurate coplanar Y and the incommensurate umbrella state in the limit of vanishing magnetic field h → 0, described in50 , identifies point E at δ = 1.1 and h = 0 as the point of the transition between the O(2) × O(2) [the incommensurate coplanar V] and the O(2) × Z2 [the incommensurate cone] breaking states. Thus point E is analogous to point D.
8 By the arguments similar to those in part (f) above, there must be an intermediate phase with broken O(2)×O(2)×Z2 . It occupies region E-C1 -a-A in Figure 5. The state between A-C1 and A-a-C1 lines is characterized by different broken symmetries (see item (b) above) which makes the line A-a-C1 to be the line of the Z3 → O(2) transition. It thus has to be viewed as a continuation of the CIT line O-A. To summarize, the quasi classical phase diagram in Figure 5 contains many different phases. It worth keeping in mind that it has been obtained under assumption of continuous phase transitions between states with different orders. Several of the shown there phase boundaries are tentative – their existence is conjectured based on different symmetry properties of the states they are supposed to separate. To highlight their conjectured nature, such lines are drawn dashed in Figure 5. These include line B-b-C2 which separates distorted umbrella (with broken Z3 × O(2) × Z2 ) and double spiral (with broken O(2) × O(2) × Z2 ), and similar to it line A-a-C1 located right below the UUD phase. The end-points of these dashed lines are conjectured to be C2 and C1 , correspondingly, which are the points of the two-magnon condensation [item (e)]. Line D-C2 , separating phases with broken O(2) × O(2) × Z2 and O(2) × Z2 , established via high-field analysis in item (f), is conjectured to end at the same C2 . Since different behavior cannot be ruled out at the present, its extension to the nearplateau region is indicated by the dashed line as well. Similar arguments apply to the line E-C1 . Finally, line C2 -C-C1 , covering the very tip of the UUD plateau phase, is made dashed because it is located past the two-magnon condensation transition (line C1 -C2 ) into the spin-nematic state instabilities of which have not been explored in sufficient details yet. One of the most unexpected and remarkable conclusions emerging from the analysis summarized here is the identification of the continuous line of C-IC transitions (line H-B-b-C2 C-C1 -a-A-O), separating phases with discrete Z3 from those with continuous O(2) symmetry. Its existence owes to the non-trivial interplay between geometric frustration and quantum spin fluctuations in the triangular antiferromagnet. Spin excitation spectra: Many of the ordered noncollinear states described above harbor spin excitations with rather unusual characteristics. It has been pointed out some time ago55,56 that local non-collinearity of the magnetic order results in the strong renormalization of the spin wave spectra at 1/S order. (This should be contrasted with the case of the collinear magnetic order, where quantum corrections to the excitation spectrum appear only at 1/S 2 order.) This interesting effect, reviewed in57 , is responsible for dramatic flattening of spin wave dispersion and/or appearance of ‘roton-like’ minima and related thermodynamic anomalies at temperatures as low as 0.1 − 0.2J 58,59 .
C.
Spin 1/2 spatially anisotropic triangular antiferromagnet with J 0 6= J
We now turn to the case of most quantum system: a spin 1/2 antiferromagnet. Qualitatively, one expects quantum fluctuations to be most pronounced in this case, which suggests,
in line with ‘order-by-disorder’ arguments of Section III A, a selection of the ordered Y, UUD and V states at and near the isotropic limit J 0 ≈ J. Behavior away from this isotropic line represents a much more difficult problem, mainly due to the absence of physically motivated small parameter, which would allow for controlled analytical calculations. Aside from the two limits where small parameters do appear, namely the high field region near the saturation field and the limit of weakly coupled spin chains (see below), the only available approach is numerical. Most of recent numerical studies of triangular lattice antiferromagnets focus on the zero field limit, h = 0, and on the phase diagram as a function of the ratio J 0 /J. These studies agree that a two-dimensional magnetic spiral order, well established at the isotropic J 0 = J point, becomes incommensurate with the lattice when J 0 6= J and persists down to approximately J 0 = 0.5J. The ordering wave vector of the spiral Q0 is strongly renormalized by quantum fluctuations54,60 away from the semiclassical result. Below about J 0 = 0.5J, strong finite size effects and limited numerical accuracy of the exact diagonalization61 and DMRG60,62 methods does not allow one to obtain a definite answer about the ground state of the spin-1/2 J − J 0 Heisenberg model.
FIG. 7: Phase diagram of the spin S = 1/2 spatially anisotropic triangular antiferromagnet in magnetic field. Vertical axis - magnetic field h/J, horizontal - dimensionless degree of spatial anisotropy, R/J = 1 − J 0 /J. Notation C/IC stands for commensurate/incommensurate phases, correspondingly. Adapted from Chen et al., Phys. Rev. B 87, 165123 (2013).
This unexpected behavior is a direct consequence of the strong frustration inherent in the triangular geometry. In the J 0 J limit the lattice decouples into a collection of linear spin chains weakly P coupled by the frustrated interchain exchange H0 = J 0 x,y Sx,y · (Sx−1/2,y+1 + Sx+1/2,y+1 ). Even classically, spin-spin correlations between spins from different chains are strongly suppressed as can be seen from the limit J 0 /J → 0 when classical spiral wave vector Q0x = 2 cos−1 [−J 0 /2J] → π + J 0 /J. In this limit the relative angle between the spin at (integer-numbered) site x of the y-th chain and its neighbor at (half-integer-numbered) x + 1/2 site of the
9 y + 1-th chain approaches π/2 + J 0 /(2J). Thus the scalar product of two classical spins at neighboring chains vanishes as J 0 /(2J) → 0. Quantum spins adds strong quantum fluctuations to this behavior, resulting in the numerically observed near-exponential decay of the inter-chain spin correlations, even for intermediate value J 0 /J . 0.560,62 . While some of the studies interpret such effective decoupling as the evidence of the spin-liquid ground state61,62 , the others conclude that the coplanar spiral ground state persists all the way to J 0 = 060,63 . Analytical renormalization group approach64 , which utilizes symmetries and algebraic correlations of low-energy degrees of freedom of individual spin-1/2 chains, finds that the system experiences quantum phase transition from the ordered spiral state to the unexpected collinear antiferromagnetic (CoAF) ground state. This novel magnetically ordered ground state is stabilized by strong quantum fluctuations of critical spin chains. This finding is supported by the coupledcluster study65,66 , functional renormalization group67 as well as by the combined DMRG and analytical RG studies in68 . It is fair to say that more studies of the very difficult J 0 /J → 0 limit of the spatially anisotropic triangular model are highly desirable in order to definitively sort out the issue of the ultimate ground state. Having described the limiting behavior along h = 0 and J 0 = J (R = 0) axes, we now discuss the full h − R phase diagram of the spin-1/2 Heisenberg model shown in Figure 7 (R = J − J 0 ). The diagram is derived from extensive DMRG study of triangular cylinders (spin tubes) composed of 3, 6 and 9 chains and of lengths 120 - 180 sites (depending on R and the magnetization value) as well as detailed analytical RG arguments applicable in the limit J 0 J 47 . It compares well, in the regions of small and intermediate R, with the variational and exact diagonalization study by Tay and Motrunich45 . The comparison is less conclusive in the regime of large anisotropy, R → 1, which is most challenging for numerical techniques as already discussed above. (The biggest uncertainty of the diagram in Figure 7 consists in so far undetermined region of stability of the cone phase and, to a lesser degree, the phase boundaries between incommensurate (IC) planar and SDW phases.) The main features of the phase diagram of the quantum spin-1/2 model are: (1) High-field incommensurate coplanar (incommensurate V or fan) phase, which is characterized by the broken O(2) × O(2) symmetry, is stable for all values of the exchange anisotropy 1 ≥ R ≥ 0. This novel analytical finding, confirmed in DMRG simulations, is described in Ref.47. This result is specific to the quantum S = 1/2 model – for any other value of the site spin S ≥ 1 there is a quantum phase transition between the incommensurate planar and the incommensurate cone phases at some RS . The critical value RS is spin-dependent and decreases monotonically with S. We find RS ≈ 0.9, 0.5, 0.4 for S = 1, 3/2, 2, respectively47 . (2) 1/3 Magnetization plateau (UUD phase) is present for all values of R too: it extends from R = 0 all the way to R = 1. This striking conclusion is based on analytical calculations near the isotropic point50 , discussed in the previous
Section, complementary field-theoretical calculations near the decoupled chains limit of R ≈ 147,69 and on extensive DMRG studies of the UUD plateau in Ref.47. (3) A large portion of the diagram in Figure 7, roughly to the right of R = 0.5, is occupied by the novel incommensurate collinear SDW phase. Physical properties of this magnetically ordered and yet intrinsically quantum state are summarized in Section IV A below. Comparing the quantum phase diagram in Figure 7 with the previously described large-S phase diagram in Figure 5, one notices that both the high-field incommensurate coplanar and the UUD phases are present there too. The fact that both of these states become more stable in the spin-1/2 case and expand to the whole range of exchange anisotropy 0 < R < 1, represents a striking quantitative difference between the largeS and S = 12 cases. It should also be noticed that both phase diagrams demonstrate that the range of stability of the incommensurate cone (umbrella) phase is greatly diminished. In contrast, a collinear SDW phase, which occupies a good portion of the quantum phase diagram in Figure 7, is not present in Figure 5 at all - and this constitutes a major qualitative distinction between the large-S and the quantum S = 12 cases.
IV.
SDW AND NEMATIC PHASES OF SPIN-1/2 MODELS A.
SDW
The collinear SDW phase is characterized by the modulated expectation value of the local magnetization hSrz i = M + Re[Φeiksdw ·r ],
(8)
where Φ is the SDW order parameter, and SDW wave vector ksdw is generally incommensurate with the lattice and, moreover, is the function of the uniform magnetization M and anisotropy R. Eq.(8) is very unusual for a classical (or semi-classical) spin system, where magnetic moments tend to behave as vectors of fixed length. It is, however, a relatively common phenomenon in itinerant electron systems with nested Fermi surfaces70–72 . The appearance of such a state in a frustrated system of coupled spin-1/2 chains is rooted in a deep similarity between Heisenberg spin chain and onedimensional spin-1/2 Dirac fermions73–75 : thanks to the wellknown phenomenon of one-dimensional spin-charge separation, the spin sectors of these two models are identical. Ultimately, it is this correspondence that is responsible for the ‘softness’ of the amplitude-like fluctuations underlying the collinear SDW state of Figure 7. Figure 8 schematically shows a dispersion of S = 1/2 electron in a magnetic field. Fermi-momenta of up- and downspin electrons k↑/↓ are shifted from the Fermi momentum of non magnetized chain, kF = π/2, by ±∆kF = ±πM , where M is the magnetization. As a result, the momentum of the spin-flip scattering processes, which determine transverse spin correlation function hS + S − i, is given by ±(k↑ − (−k↓ )) = ±2kF = ±π, and remains commensurate
10 with the lattice. At the same time, longitudinal spin excitations, which preserve S z , now involve momenta ±2k↑/↓ = π(1 ± 2M ) and become incommensurate with the lattice. To put it differently, in the magnetized chain with M 6= 0 low-energy longitudinal spin fluctuations can be parameterized as S z (x) ∼ S z ei(π−2πM )x + S z∗ ei(−π+2πM )x , where (calligraphic) S z (x) represents slow (low-energy) longitudinal mode. Similarly, transverse spin fluctuations are written as S + (x) ∼ S + eiπx , with S + representing low-energy transverse mode.
FIG. 8: Schematics of spinon dispersion for one-dimensional spin chain in magnetic field. Dashed line shows dispersion for zero field. k↑ (k↓ ) denote Fermi momentum of spin ’up’ (’down’) spinons correspondingly. Fermi momentum in the absence of the field is kF = π/2.
This simple fact has dramatic P consequences for frustrated interchain interaction H0 = J 0 x,y Sx,y · (Sx−1/2,y+1 + Sx+1/2,y+1 ). Longitudinal (z) component of the sum of two z neighboring spins on chain (y + 1) adds up to Sx−1/2,y+1 + z z i(π−2πM )x Sx+1/2,y+1 → sin[πM ](Sx,y+1 e + h.c.), while the sum of their transverse components becomes a derivative of the smooth component of the transverse field S + , + + + Sx−1/2,y+1 + Sx+1/2,y+1 → eiπx ∂x Sx,y+1 . Hence the lowenergy interaction reduces to H0 → P R limit0 of the inter-chain + 0 − z∗ z S dx{J sin[πM ]S x,y x,y+1 + J Sx,y ∂x Sx,y+1 + h.c.}. y The presence of the spatial derivative in the second term severely weakens it69 and results in the domination of the density-density interaction (first term) over the transverse one (second term). The field-induced shift of the Fermi-momenta from its commensurate value, kF → k↑/↓ , together with frustrated geometry of inter-chain exchanges, are the key reasons for the field-induced stabilization of the two-dimensional longitudinal SDW state. Symmetry-wise, the SDW state breaks no global symmetries (time reversal symmetry is broken by the magnetic field, which also selects the z axis), and, in particular, it preserves U (1) symmetry of rotations about the field axis. This crucial feature implies the absence of the off-diagonal magnetic or-
der hSrx,y i = 0 and would be gapless (Goldstone) spin waves. Instead, SDW breaks lattice translational symmetry. Its order parameter Φ ∼ hS z i 6= 0 is determined by inter-chain interactions69,76 . Provided that ksdw = π(1 − 2M )ˆ x is incommensurate with the lattice, the only low energy mode is expected to be the pseudo-Goldstone acoustic mode of broken translations, known as a phason. (Inter-chain interactions do affect ksdw but in the limit of small J 0 /J this can be neglected.) The phason is a purely longitudinal mode corresponding to the phase of the complex order parameter Φ and hence represents a modulation of S z only. This too is unusual in the context of insulating magnets, where, typically, the low energy collective modes are transverse spin waves, associated with small rotations of the spins away from their ordered axes. In the spin wave theory, longitudinal modes are typically expected to be highly damped57,77,78 , and hence hard to observe. (For a recent notable exception to this rule see a study of amplitude-modulated magnetic state of PrNi2 Si2 79 .) In the SDW state, the longitudinal phason mode in the only low energy excitation. Transverse spin excitations, which SDW also supports, have a finite spectral gap. This, in fact, is one of key experimentally identifiable features of the SDW phase. More detailed description of spin excitations of this novel phase, as well as of the spin-nematic state reviewed below, can be found in the recent study76 . At present, there are three known routes to the field-induced longitudinal SDW phase for a quasi-one-dimensional system of weakly coupled spin chains. The first, reviewed above, relies on the geometry-driven frustration of the transverse inter-chain exchange, which disrupts usual transverse spin ordering and promotes incommensurate order of longitudinal S z components. The other route, described in the subsection IV A 1 below, relies on Ising anisotropy of individual chains. Lastly, it turns out that a two-dimensional SDW state may also emerge in a system of weakly coupled nematic spin chains - this unexpected possibility is reviewed in the subsection IV B.
1.
SDW in a system of Ising-like coupled chains
There is yet another surprisingly simple route to the twodimensional SDW phase. It consists in replacing Heisenberg chains with XXZ ones with pronounced Ising anisotropy. It turns out that a sufficiently strong magnetic field, applied along the z (easy) axis, drives individual chains into a critical Luttinger liquid state with dominant longitudinal, S z − S z , correlations80 . This crucial property ensures that weak residual inter-chain interaction selects incommensurate longitudinal SDW state as the ground state of the anisotropic twodimensional system. We note, for completeness, that not every field-induced gapless spin state is characterized by the dominant longitudinal spin correlations. For example, another well-known gapped system, spin-1 Haldane chain, can too be driven into a critical Luttinger phase by sufficiently strong magnetic field81–84 . However, that critical phase is instead dominated by strong transverse spin correlations85,86 . As a result, a 2d
11 ground state of weakly coupled spin-1 chains is a usual cone state87 .
2.
Magnetization plateau as a commensurate collinear SDW phase
The commensurate case, when the wavelength λsdw = 2π/ksdw is a rational fraction of the lattice period, λsdw = q/p, requires special consideration. (Here the lattice period is set to be 1 and q and p are integer numbers.) Such commensurate state is possible at commensurate magnetization values M (p,q) = 12 (1 − 2q p ). At these values, ‘sliding’ SDW state locks-in with the lattice, resulting in the loss of continuous translational symmetry. The SDW-plateau transition is then an incommensurate-commensurate transition of the sine-Gordon variety47,69 . It turns out that in two-dimensional triangular lattice such locking is possible, provided that integers p and q have the same parity (both even or both odd)69 (and, of course, provided that the spin system is in the two-dimensional collinear SDW phase). This condition selects M = 1/3 Msat plateau (q = 1, p = 3) as the most stable one, in a sense of the biggest energy gap with respect to creation of spin-flip excitation (which changes total magnetization of the system by ±1). The next possible plateau is at M = 3/5 Msat (q = 1, p = 5)69 – however this one apparently does not realize in the phase diagram in Figure 7, perhaps because it is too narrow and/or happen to lie inside the (yet not determined numerically) cone phase. Applied to the one-dimensional spin chain, the above condition can be re-written as a particular S = 21 version of the Oshikawa-Yamanaka-Affleck condition88,89 for the period-p magnetization plateau in a spin-S chain, pS(1 − M/Msat ) = integer. Interestingly, this shows that p = 3 plateau at M = 1/3 Msat of the total magnetization Msat is possible for all values of the spin S: the quantization condition becomes simply 2S = integer. This rather non-obvious feature has in fact been numerically confirmed in several extensive studies90–92 . B.
condensation hSr− Sr−0 i = hQ−− i = 6 0. As in a superconductor, a two-magnon condensate breaks U (1) symmetry, which in this case is a breaking of the spin rotational symmetry with respect to magnetic field direction. It does not, however, break time-reversal symmetry (which requires a single-particle condensation). Just as in a superconductor, single-particle excitations of the nematic phase are gapped. This implies that transverse spin correlation function hSr+ S0− i ∼ e−r/ξ , which probes single magnon excitations, is short-ranged and decays exponentially. At the same time, fluctuations of magnon density, which are probed by longitudinal spin correlation function hSrz S0z i, are sound-like acoustic (Bogoliubov) modes. 1.
Weakly coupled nematic chains
Basic ingredients of this picture - gapped magnon excitations and attractive interaction between them - are nicely realized in the spin-1/2 quasi-one-dimensional material LiCuVO4 , reviewed in Section V B: the gap in the magnon spectrum is caused by the strong external magnetic field h (1) which exceeds (single-particle) condensation field hsat , while the attraction between magnons is caused by the ferromagnetic (negative) sign of exchange interaction J1 between the nearest spins of the chain. Under these conditions, the twomagnon bound state, which lies below the gapped single par(2) (1) ticle states, condenses at hsat , which is higher than hsat . As a result, a spin nematic state is naturally realized in each in(1) dividual chain in the intermediate field interval hsat < h < (2) 99 hsat .
Spin nematic
Spin nematic represents another long-sought type of exotic ordering, Out of many possible nematic states93,94 , our focus here is on bond-nematic order associated with the twomagnon pairing95 and the appearance of the non-local order parameter Q−− = Sr− Sr−0 defined on the hr, r0 i bond connecting sites r and r0 . Such order parameter can be build from quadrupolar operators Qx2 −y2 = Srx Srx0 − Sry Sry0 and Qxy = Srx Sry0 + Sry Srx0 , as Q−− = Qx2 −y2 − iQxy 96 . This bond-nematic order is possible in both S ≥ 1, where quadrupolar order was originally suggested97 , and S = 1/2 systems of localized spins, coupled by exchange interaction. The magnon pairing viewpoint, explored in great length in96,98,99 , is extremely useful for understanding basic properties of the spin-nematic state: the nematic can be thought of as a ‘bosonic superconductor’ formed as a result of two-magnon
FIG. 9: (a) Weakly coupled nematic chains with ferromagnetic J1 (solid lines) and antiferromagnetic next-nearest J2 , coupled by an inter-chain exchange J 0 . (b) Same system viewed as a set of weakly coupled zig-zag chains.
Note, however, that a true U (1) symmetry breaking is not possible in a single chain, where instead a critical Luttinger state with algebraically decaying nematic correlations
12 is established96 . To obtains a true two-dimensional nematic phase, one needs to establish a phase coherence between the phases of order parameters Q−− (y) of different chains. By our superconducting analogy, this requires a Josephson coupling to transfer (hop) bound two-magnon pairs between nematic chains. The corresponding ‘hopping’ term reads P K x,y (Q++ (x, y)Q−− (x, y + 1) + h.c.). Microscopically, such an interaction represents a four-spin coupling, which is not expected to be particularly large in a good Mott insulator with a large charge gap, such as LiCuVO4 . However, even if K is absent microscopically, it will be generated from the usual inter-chain spin exchange Pperturbatively − + J 0 x,y (Sx,y Sx,y+1 + h.c.), which plays the role of a singleparticle tunneling process in the superconducting analogy. (Observe that expectation value of this interaction in the chain nematic ground state is zero – adding or removing of a single magnon to the ‘superconducting magnon’ ground state is forbidden at energies below the single magnon gap.) The pairtunneling generated by fluctuations is estimated to be of the order K ∼ (J 0 )2 /J1 J 0 . At the same time, S z − S z interaction between chains does not suffer from a similar ‘low-energy suppression’. This is because S z is simply proportional to a number of magnon pairs, npair (x, y) = 2n(x, y), which is just twice the magnon number n(x, y). Hence S z (x, y) = 1/2 − 2n(x, y) differs only a by coefficient 2 from its usual expression in terms of magnon density. We thus have a situation where the strength of interchain density-density (S z − S z ) interaction, which is determined by the original interchain J 0 , is much stronger than that for the fluctuation-generated Josephson interaction K ∼ (J 0 )2 /J1 . In addition, more technical analysis of the scaling dimensions of the corresponding operators shows76 , that the two competing interactions are characterized by (almost) the same scaling dimension (approximately equal to 1 for h ≈ hsat ) which makes them both strongly relevant in the renormalization group sense. Given an inequality J 0 (J 0 )2 /J1 , which selects interchain S z − S z interaction as the strongest one, we end up with a two-dimensional collinear SDW phase build out of nematic spin chains76,100 . This conclusion holds for all h (2) except for the immediate vicinity of the saturation field hsat . There a separate fully two-dimensional BEC analysis is required, due to the vanishing of spin velocity at the saturation field, and the result is a true two-dimensional nematic phase (1) (2) in the narrow field range hsat . h ≤ hsat 76,99,100 . A useful analogy to this competition is provided by models of striped superconductors, where the competition is between the superconducting order (a magnetic analogue of which is the spin nematic) and the charge-density wave order (a magnetic analogue of which is the collinear SDW), see Ref.101 and references therein. To summarize, weak inter-chain interaction J 0 between J1 − J2 spin chains with strong nematic spin correlations actually stabilizes a two-dimensional SDW phase as the ground state in a wide range of magnetization. This state preserves U (1) symmetry of spin rotations and is characterized by shortranged transverse spin correlations, similar to a nematic state.
2.
Spin-current nematic state at the 1/3-magnetization plateau
The discussion in the previous Subsection was focused on the systems with ferromagnetic exchange (J1 < 0) on some of the bonds - as described there, negative exchange needed in order to obtain an attractive interaction between magnons. Superconducting analogy, extensively used above, forces one to ask, by analogy with superconducting states of repulsive fermion systems (such as, for example, pnictide superconductors or high-temperature cuprate ones), if it is possible to realize a spin-nematic in a spin system with only antiferromagnetic (that is, repulsive) exchange interactions between magnons. To the best of our knowledge, the first example of such a state is provided by the spin-current state described in part (e) of the Section III B. Being nematic, this state is characterized by a spin current long-range order and the absence of the magnetic long-range order in the transverse to the magnetic field direction51 . A very similar state, named chiral Mott insulator, was recently discovered in the variational wave function study of a two-dimensional system of interacting bosons on frustrated triangular lattice102 as well as in a one-dimensional system of bosons on frustrated ladder103,104 . In both cases, a chiral Mott insulator is an intermediate phase, which separates the usual Mott insulator state (which is a boson’s analogue of the UUD state) from the superfluid one (which is an analogue of the cone state). As in Figure 5, it intervenes between the states with distinct broken symmetries (Z3 and O(2) × Z2 in our case), and gives rise to two continuous transitions instead of a single discontinuous one.
C.
Magnetization plateaus in itinerant electron systems
Up-up-down magnetization plateaus, found in the triangular geometry, are of classical nature. Over years, several interesting suggestions of non-classical (liquid-like) magnetization plateaux have been put forward105–110 , but so far not observed in experiments or numerical simulations. Very recently, however, two numerical studies111,112 of the spin-1/2 kagom´e antiferromagnet have observed non-classical magnetization plateaus at M/Msat = 1/3, 5/9, 7/9. These intriguing findings, taken together with earlier prediction of a collinear spin liquid at h = hsat /3 in the classical kagom´e antiferromagnet113 , hint at a very rich magnetization process of the quantum model, the ground state of which at h = 0 is a Z2 spin liquid!4 . A different point of view on the magnetization plateau was presented recently in Ref.114. The authors asked if the plateau is possible in an itinerant system of weakly-interacting electrons. The answer to this question is affirmative, as can be understood from the following consideration. Let us start with a system of non-interacting electrons on a triangular lattice. A magnetic field, applied in-plane in order to avoid complications due to orbital effects, produces magnetization M = (n↑ − n↓ )/2, where densities nσ of electrons with spin σ =↑, ↓ are constrained by the total density n = n↑ + n↓ . Consider now special situation with n↑ = 3/4,
13 at which the Fermi-surface of σ =↑ electrons, by virtue of lattice geometry, acquires particularly symmetric shape: a hexagon inscribed inside the Brillouin zone hexagon, see Figure 10.
FIG. 10: The Fermi surface (red hexagon) of the non-interacting σ =↑ electrons on a triangular lattice at n↑ = 3/4. Nearly circular Fermi surface of the minority σ =↓ electrons is shown by blue line. Bold black hexagon represents the Brillouin zone. Adapted from Zhihao Hao and Oleg A. Starykh, Phys. Rev. B 87, 161109 (2013).
Points where σ =↑ Fermi-surface touches the Brillouin zone (denoted by vectors ±Qj with j = 1, 2, 3 in Fig.10) are the van Hove points, at which Fermi-velocity vanishes and electron dispersion becomes quadratic. They are characterized by the logarithmically divergent density of states. In addition, being a hexagon, σ =↑ Fermi-surface is perfectly nested. As a result, static susceptibility χ↑ (k) of spin-up electrons is strongly divergent, as log2 (|k−Qj |), for wave vectors k ≈ Qj . Given this highly susceptible spin-↑ Fermi surface, it is not surprising that a weak interaction between electrons, either in the form of a direct density-density interaction V nr nr+δj between electrons on, e.g., nearest sites, or in the form of a local Hubbard interaction U nr,↑ nr,↓ between the majority and minority particles, drives spin-↑ electrons into a gapped correlated state - the charge density wave (CDW) state114 with fully gapped Fermi surface. Moreover, CDW ordering wave vectors Qj are commensurate with the lattice, leading to a commensurate CDW for the spin-↑ electrons115 . Minority spin-↓ electrons experience position-dependent effective field U hnr,↑ i and form CDW as well. Depending on whether or not vectors Qj span the spin-down Fermi-surface (this depends on n↓ density), the Fermi-surface of σ =↓ electrons may or may not experience reconstruction. However, being not nested, it is guaranteed to retain at least some parts of the critical Fermi-surface. The resulting state is a co-existence of a charge- and collinear spin-density waves, together with critical σ =↓ Fermi surface. Since the energy cost of promoting a spin-↓ electron to a spin-↑ state is finite (and given by the gap on the
spin-↑ Fermi surface), the resulting state realizes fractional magnetization plateau, the magnetization of which is determined by the total density n via M = (3/2 − n)/2. At halffilling, n = 1, the plateau is at 1/2 of the total magnetization, but for n 6= 1 it takes a fractional value. Amazingly, the obtained state is also a half-metal116 - the only conducting band is that of (not gapped) minority spin-↓ electrons. Theoretical analysis sketched here bears strong similarities with recent proposals115,117–120 of collinear and chiral spindensity wave (SDW) and superconducting states of itinerant electrons on a honeycomb lattice in the vicinity of electron filling factors 3/8 and 5/8 at zero magnetization. Our analysis shows that even simple square lattice may host similar half-metallic magnetization plateau state, see supplement to114 . Similar to the case of a magnetic insulator, described in the previous Sections, external magnetic field sets the direction of the collinear CDW/SDW state. The resulting halfmetallic state only breaks the discrete translational symmetry of the lattice, resulting in fully gapped excitations, and remains stable to fluctuations of the order parameter about its mean-field value. In addition to standard solid state settings, the described phenomenon may also be observed in experiments on cold atoms, where desired high degree of polarization can be easily achieved121 . It appears that, in addition to the half-metallic state, the system may also support p-wave superconductivity - a competition between these phases may be efficiently studied with the help of functional renormalization group122 .
V.
EXPERIMENTS
Much of the current theoretical interest in quantum antiferromagnetism comes from the amazing experimental progress in this area during the last decade. The number of interesting materials is too large to review here, and for this reason we focus on a smaller sub-set of recently synthesized quantum spin-1/2 antiferromagnets, which realize some of quantum states discussed above. One of the best known among this new generation of materials is Cs2 CuCl4 , extensively studied by Coldea and collaborators in a series of neutron scattering experiments123–125 and by others via NMR126 and, more recently, ESR127–130 experiments. This spin-1/2 material represents a realization of a deformed triangular lattice with J 0 /J = 0.34124 and significant DM interactions on chain and inter-chain (zig-zag) bonds, connecting neighboring spins69,124 . Inelastic neutron scattering experiments have revealed unusually strong multi-particle continuum, the origin of which has sparked intense theoretical debate131–138 . The current consensus is that Cs2 CuCl4 is best understood as a weakly-ordered quasi-one-dimensional antiferromagnet, whose spin excitations smoothly interpolate from fractionalized spin-1/2 spinons of one-dimensional chain at high- and intermediate energies to spin waves at lowest energy ( J 0 )138 . Although weak, residual inter-plane and DM interactions play the dominant role in the magnetization process of this material. The resulting B − T phase diagram is rather complex and highly anisotropic139 , and does not con-
14 tain a magnetization plateau. However it is worth mentioning that this was perhaps the first spin-1/2 material, a magnetic response of which featured a SDW-like phase ordering wave vector, which scales linearly with magnetic field in an about 1 Tesla wide interval (denoted as phase “S” in123 and as phase “E” in139 ). While still not well understood, this experimental observation have provided valuable hint to quasi-1d approach based on viewing Cs2 CuCl4 as a collection of weakly coupled spin chains138 . A.
Magnetization plateau
Robust 1/3 magnetization plateau – the first of its kind among triangular spin-1/2 antiferromagnets – is present in Cs2 CuBr4 , which has the same crystal structure as Cs2 CuCl4 , but is less deformed, J 0 /J ≈ 0.7, and is more twodimensional than the chloride-based material. The observed plateau, which is about 1 Tesla wide (hc1 = 13.1T and hc2 = 14.4T)140–142 , is clearly visible in both magnetization and elastic neutron scattering measurements141,142 , which determined the UUD spin structure on the plateau. The observation of the magnetization plateau has generated a lot of experimental activity. The quantum origin of the plateau visibly manifests itself via essentially temperature-independent plateau’s critical fields hc1,2 (T ) ≈ hc1,2 (T = 0), as found in the thermodynamic study143 . This behavior should be contrasted with the phase diagram of the spin-5/2 antiferromagnet RbFe(MoO4 )2 27,144,145 , where the critical field hc1 (T ) does show strong downward shift with T . (Recall that in the classical model, Figure 3, the UUD phase collapses to a single point at T = 0.) 1.4 1/3 uud 1.2 A
Paramagnetic
Temperature (K)
1.0 B 0.8
I
2/3
IIa
0.6
IIb
0.4
III
IV
Hs
0.2
0.0 10
V
15
20
25
30
Field (tesla)
FIG. 11: Magnetic phase diagram of Cs2 CuBr4 , as deduced from the magnetocaloric-effect data taken at various temperatures. Circles indicate second-order phase boundaries, whereas other symbols except the open diamonds indicate first-order boundaries. Adapted from Fortune et al., Phys. Rev. Lett. 102, 257201 (2009).
Commensurate up-up-down spin structure of Cs2 CuBr4 is also supported by NMR measurements146,147 which in addition finds that transitions between the commensurate plateau and adjacent to it incommensurate phases are discontinuous (first-order). Extensive magnetocaloric effect and magnetictorque experiments148 have uncovered surprising cascade of
field-induced phase transitions in the interval 10 − 30 T. The most striking feature of the emerging complex phase diagram is that it appears to contain up to 9 different magnetic phases – in stark contrast with the ‘minimal’ theoretical model diagram in Figure 3 which contains just 3 phases! This, as well as strong sensitivity of the magnetization curve to the direction of the external magnetic field with respect to crystal axis, strongly suggest that the difference in the phase diagrams has to do with spatial (J 0 6= J) and spin-space (asymmetric DM interaction) anisotropies present in Cs2 CuBr4 . Large-S and classical Monte Carlo studies28 do find the appearance of new incommensurate phases in the phase diagram, in qualitative agreement with the large-S diagram of Figure 5 (note that the latter does not account for the DM interaction which significantly complicates the overall picture28 ). Perhaps the most puzzling of the “six additional” phases is a narrow region at about B = 23T, where dM/dB exhibits sharp double peak structure, interpreted in142,149 as a novel magnetization plateau at M/Msat = 2/3. Such a new 2/3-magnetization plateau was observed in an exact diagonalization study of spatially anisotropic spin-1/2 model150 but was not seen in more recent variational wave function45 and DMRG47 , as well as in analytical large-S 28,50 studies. Nearly isotropic, J 0 /J ≈ 1, antiferromagnet Ba3 CoSb2 O9 is believed to provide an ‘ideal’ realization of the spin-1/2 antiferromagnet on a uniform triangular lattice151,152 . And, indeed, its experimental phase diagram is in close correspondence with J 0 = J ‘cut in Figure 5 (along δ = 0 line) and Figure 7 (along R = 0 line): it has 120◦ spin structure at zero field, coplanar Y state at low fields, the 1/3 magnetization plateau in the hc1 /hsat = 0.3 ≤ h/hsat ≤ hc2 /hsat = 0.47 interval, and coplanar V state (denoted as 2 : 1 state in151 ) at higher fields. A new element of the study151,152 is the appearance of weak anomaly in dM/dB at about M/Msat = 3/5, which was interpreted as a quantum phase transition from the V phase to another coplanar phase - inverted Y (state ‘e’ in Figure 2). Near the saturation field these two phases are very close in energy, the difference appears only in the 6th order in condensate amplitude53 . Such a transition can be driven by sufficiently strong easy-plane anisotropy153 as well as anisotropic DM interaction28 . Perhaps, the more relevant to Ba3 CoSb2 O9 is another possibility - that a transition is driven by the interlayer interaction. Ref. 154 has shown that weak inter-plane antiferromagnetic exchange interaction causes transition from the uniform V phase to the staggered V phase. The latter is described by the same Eq.(5) but with a z-dependent phase, ϕz = ϕ˜ + πz (here, z is the integer coordinate of the triangular layer and ϕ˜ is an overall constant phase), leading to the doubling of the period of the magnetic structure along the direction normal to the layer. It is easy to see that such a state actually gains energy from the antiferromagnetic interlayer exchange J 00 , while preserving the optimal in-plane configuration in every layer. Such a transition, denoted as HFC1-HFC2 transition, was also observed in recent semi-classical Monte Carlo simulations155 . This development suggest that a mysterious ‘2/3-plateau’ of Cs2 CuBr4 , mentioned above, may too
15 be related to a transition between the lower-field uniform and higher-field staggered versions of the commensurate V phase.
B.
SDW and spin nematic phases
A collinear SDW order has been observed in spin-1/2 Ising-like antiferromagnet BaCo2 V2 O8 . Experimental confirmations of this comes from specific heat156 and neutron diffraction157 measurements. The latter one is particularly important as it proofs the linear scaling of the SDW ordering wave vector with the magnetization, ksdw = π(1 − 2M ), predicted in158 . Subsequent NMR159 , ultrasound160 , and neutron scattering161 experiments have refined the phase diagram and even proposed the existence of two different SDW phases159 stabilized by competing interchain interactions. Most recently, spin-1/2 magnetic insulator LiCuVO4 has emerged162,163 as a promising candidate to realize both a highfield spin nematic phase, right below the two-magnon saturation field, which is about 45 T high, and an incommensurate collinear SDW phase at lower fields, extending from about 40T down to about 10 T. At yet lower magnetic field, the material realizes more conventional vector chiral (umbrella) state which can be stabilized by a moderate easy-plane anisotropy of exchange interactions164 (which does not affect the high field physics discussed here). This last material seems to nicely realize theoretical scenario outlined in Section IV B 1: spin-nematic chains165,166 form a two-dimensional nematic phase only in the immediate vicinity of the saturation field167 . At fields below that rather narrow interval, the ground state is an incommensurate longitudinal SDW state. Evidence for the latter includes detailed studies of NMR line shape168–171 and neutron scattering172,173 . It is worth adding here that quasi-one-dimensional nature of this material is evident from the very pronounced multi-spinon continuum, observed at h = 0 in inelastic neutron scattering studies174 .
C.
Weak Mott insulators: Hubbard model on anisotropic triangular lattice
Given that, quite generally, Heisenberg Hamiltonian can be viewed as a strong-coupling (large U/t) limit of the Hubbard model, it is natural to consider the fate of the Hubbard t − t0 − U model on (spatially anisotropic, in general) triangular lattice. As a matter of fact, this very problem is of immediate relevance to intriguing experiments on organic Mott insulators of X[Pd(dmit)2 ]2 and κ-(ET)2 Z families. Recent experimental8,175 and theoretical12,176–178 reviews describe key relevant to these materials issues, and we direct interested readers to these publications. One of the main unresolved issues in this field is that of a proper minimal model that captures all relevant degrees of freedom. Highly successful initial proposal181 models the system as a simple half-filled Hubbard model on a spatially
FIG. 12: (a) Spatially anisotropic single-band Hubbard model for organic Mott insulators, after Refs.8,179. Note that t0 /t < 1 here implies J/J 0 < 1 in Figure 1: J 0 of Fig. 1 actually lives on t bonds of the Hubbard model here. (b) Geometry of an extended two-band model, after Ref.180. The ’sites’, shown by filled dots inside ovals representing dimers, are now two-molecule dimers. Two sites from the same dimer are connected by the intra-dimer tunneling amplitude td , and in the limit td ’inter-dimer tunnelings’ the model reduces to that in (a). Note also the appearance of additional hopping amplitudes (dotted line) connecting ’more distant’ sites of neighboring dimers.
anisotropic triangular lattice, as sketched in Figure 12. Every ‘site’ of this lattice in fact represents closely bound dimer, made of two ET molecules181 , and occupied by single electron (or hole). Spatial anisotropy shows up via different hopping integrals t, t0 . Within the standard large-U description, anisotropy of hopping t0 /t directly translates into that of exchange interactions on different bonds, J 0 /J ∼ (t0 /t)2 . Ironically, most of the studied materials fall onto t0 < t side8,179 of the diagram, which happens to be opposite to J > J 0 limit of spatially anisotropic Heisenberg model, to which this review is devoted. This description has generated a large number of interesting studies, the full list of which is beyond the scope of this review. One of the main outcomes of these studies is the establishment of approximate t0 /t − U phase diagram (see for example Figure 6 of182 ) which harbors metallic phase (for U/t . 10 or less, depending on t0 /t) and various insulating magnetic phases, which include both the standard N´eel and non-coplanar spiral phases as well as quantum-disordered spin-liquid state (for t0 /t ≈ 0.9 and U/t & 12). Thinking in terms of effective spin-only model, it is important to realize that for not too large U/t, the standard Heisenberg model must be amended with ring-exchange terms involving four (or more) long spin loops183,184 . This addition dramatically affects the regime of intermediate U/t by stabilizing an insulating spin-liquid ground state184,185 . The nature of the emerging spin-liquid is subject of intense on-going investigations, with proposals ranging from Z2 liquid186,187 to spin Bose-metal188 , to spin-liquid with quadratic band touching189 . Recently, however, this appealing spin-only picture of the organic Mott insulators has been challenged by the experimental discovery of anomalous response of dielectric constant190 and lattice expansion coefficient191 at low temperature. This finding imply that charge degrees of freedom, assumed frozen in the spin-only description, are actually present
16 in the material and have to be accounted for in theoretical modeling. Several subsequent papers180,192,193 have identified dimer units of the triangular lattice, viewed as sites in Figure 12, as the most likely place where charge dynamics persists down to lowest temperatures. To describe these internal states of the two-molecule dimers, one need to go back to a two-band extended Hubbard model description194 . Taking the strong-coupling of such a model, one derives180 a coupled dynamics of interacting spins and electric dipoles on the triangular lattice. In turns out that sufficiently strong inter-dimer Coulomb interaction stabilizes charge-ordered state (dipolar solid) and suppresses spin ordering via non-trivial modification of exchange interactions J, J 0 . Clearly many more studies, both experimental and theoretical, are required in order to elucidate the physics behind apparent spin-liquid behavior of organic Mott insulators. In place of conclusion we just state the obvious: despite many years of investigations, quantum magnets on triangular lattices continue to surprise us. There are no doubts that future studies of new materials and models, inspired by them, will bring out new quantum states and phenomena.
collaborations and countless insightful discussions that provide the foundation of this review. I thank Sasha Abanov, Hosho Katsura, Ru Chen, Hyejin Ju, Hong-Chen Jiang, Christian Griset, and Shane Head for their crucial contributions to joint investigations related to the topics discussed here. Discussions of experiments with Collin Broholm, Radu Coldea, Martin Mourigal, Masashi Takigawa, Leonid Svistov, Alexander Smirnov and Yasu Takano are greatly appreciated as well. I have benefited extensively from conversations with Cristian Batista, Misha Raikh, Dima Pesin, Eugene Mishchenko and Oleg Tchernyhyov. Many thanks to Luis Seabra, Nic Shannon, Alexander Smirnov and Yasu Takano for permissions to reproduce figures from their papers in this review. Special thanks to Andrey Chubukov for the critical reading of the manuscript and invaluable comments. This work is supported by the National Science Foundation through grant DMR-1206774.
Acknowledgments
I am grateful to my friends and coauthors - Leon Balents, Andrey Chubukov, Jason Alicea and Zhihao Hao - for fruitful
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